I've come up with this question when reading Shoichiro Sakai's book $C^{*}$-Algebras and $W^{*}$-algebras Chapter 1.8.
Let $A$ be a $W^{*}$-algebra (a $C^{*}$-algebra with a Banach space predual) and $V$ a norm-dense subspace of the predual $A_{*}\subseteq A^{*}$ of $A$. $V$ is assumed to be invariant in the sense that for each $a\in A$ and $\varphi\in V$, all of $(x\mapsto\varphi(ax))$, $(x\mapsto\varphi(xa))$, and $(x\mapsto\overline{\varphi(x^{*})})$ are still in $V$.
For example, if $A=\mathcal{B}(H)$ for some Hilbert space $H$, then we might choose $V$ to be the set of finite-rank operators on $H$, which is a dense subspace of $\mathcal{B}(H)_{*}$, the set of trace-class operators on $H$. Or, $A_{*}$ itself is an example of invariant subspace.
We can define the $V$-strong topology on $A$ as the locally convex topology generated by the set of seminorms given by $$x\mapsto\varphi(x^{*}x)^{1/2}$$ for each positive linear functional $\varphi\in V$. Let us denote this topology as $s(A,V)$.
Question: Is $s(A,V)$ a dual topology for the pairing $(A,V)$? That is, is the following true? $$\sigma(A,V)\subseteq s(A,V)\subseteq\tau(A,V)$$
I believe I could show the followings:
For $A=\mathcal{B}(H)$ and $V=H^{*}\otimes H$ the set of finite rank operators, then $s(A,V)$ is the usual strong operator topology. In this case, the above claim is true.
$\sigma(A,V)\subseteq s(A,V)\subseteq\tau(A,A_{*})$. Hence, the dual of $(A,s(A,V))$ is a subspace of $A_{*}$ containing $V$.
If $B$ is the unit ball in $A$, then $\mathbf{1}_{B}:(B,\tau(A,V))\rightarrow(B,s(A,V))$ is continuous.
Edit
Here is the precise definition of $\sigma(A,V)$ and $\tau(A,V)$.
$\sigma(A,V)$ is the coarsest topology making each element in $V$ a continuous linear functional on $A$. More concretely, a neighborhood base at $0$ is given as $$\left\{a\in A:|\varphi(a)|<\epsilon\right\}$$ for $\varphi\in V$ and $\epsilon>0$.
$\tau(A,V)$ is the Mackey topology of the pairing $(A,V)$. More concretely, a neighborhood base at $0$ is given as $$\left\{a\in A:\sup_{\varphi\in K}|\varphi(a)|<\epsilon\right\}$$ for $\sigma(V,A)$-compact absolutely convex subset $K\subseteq V$ and $\epsilon>0$. In other words, $K$ is a weakly compact absolutely convex subset of the Banach space $A_{*}$, and at the same time $K$ is contained in $V$.